scholarly journals Contractions of Representations and Algebraic Families of Harish-Chandra Modules

2018 ◽  
Vol 2020 (11) ◽  
pp. 3494-3520 ◽  
Author(s):  
Joseph Bernstein ◽  
Nigel Higson ◽  
Eyal Subag
Keyword(s):  
Lie Groups ◽  
Unitary Group ◽  
Reductive Group ◽  
Mathematical Physics ◽  
Point Of View ◽  
Unitary Representations ◽  
Group Representations ◽  
Algebraic Point ◽  
Infinite Dimensional ◽  
Real Forms

Abstract We examine from an algebraic point of view some families of unitary group representations that arise in mathematical physics and are associated to contraction families of Lie groups. The contraction families of groups relate different real forms of a reductive group and are continuously parametrized, but the unitary representations are defined over a parameter subspace that includes both discrete and continuous parts. Both finite- and infinite-dimensional representations can occur, even within the same family. We shall study the simplest nontrivial examples and use the concepts of algebraic families of Harish-Chandra pairs and Harish-Chandra modules, introduced in a previous paper, together with the Jantzen filtration, to construct these families of unitary representations algebraically.

Elements of Group Theory

2021 ◽  
pp. 51-110
Author(s):  
J. Iliopoulos ◽  
T.N. Tomaras
Keyword(s):  
Group Theory ◽  
Lie Groups ◽  
Lorentz Group ◽  
Unitary Representations ◽  
Compact Groups ◽  
Mathematical Language ◽  
Group Representations ◽  
Infinite Groups ◽  
Infinite Dimensional ◽  
Symmetry Properties

The mathematical language which encodes the symmetry properties in physics is group theory. In this chapter we recall the main results. We introduce the concepts of finite and infinite groups, that of group representations and the Clebsch–Gordan decomposition. We study, in particular, Lie groups and Lie algebras and give the Cartan classification. Some simple examples include the groups U(1), SU(2) – and its connection to O(3) – and SU(3). We use the method of Young tableaux in order to find the properties of products of irreducible representations. Among the non-compact groups we focus on the Lorentz group, its relation with O(4) and SL(2,C), and its representations. We construct the space of physical states using the infinite-dimensional unitary representations of the Poincaré group.

scholarly journals Infinite-Dimensional Lie Groups and Algebras in Mathematical Physics

2010 ◽  
Vol 2010 ◽  
pp. 1-35 ◽  
Author(s):  
Rudolf Schmid
Keyword(s):  
Lie Groups ◽  
Mathematical Physics ◽  
Quantum Field ◽  
Loop Groups ◽  
Diffeomorphism Groups ◽  
Gauge Groups ◽  
Infinite Dimensional ◽  
Lie Groups And Algebras ◽  
Volume Preserving ◽  
Infinite Dimensional Lie Groups

We give a review of infinite-dimensional Lie groups and algebras and show some applications and examples in mathematical physics. This includes diffeomorphism groups and their natural subgroups like volume-preserving and symplectic transformations, as well as gauge groups and loop groups. Applications include fluid dynamics, Maxwell's equations, and plasma physics. We discuss applications in quantum field theory and relativity (gravity) including BRST and supersymmetries.

GEOMETRY OF UNIVERSAL GRASSMANN MANIFOLD FROM ALGEBRAIC POINT OF VIEW

1989 ◽  
Vol 01 (01) ◽  
pp. 1-46 ◽  
Author(s):  
KANEHISA TAKASAKI
Keyword(s):  
Lie Algebra ◽  
Grassmann Manifold ◽  
Point Of View ◽  
Simple Application ◽  
Grassmann Manifolds ◽  
Algebraic Point ◽  
Infinite Dimensional ◽  
Component Theory ◽  
Infinitesimal Transformations ◽  
Algebraic Formulation

An algebraic formulation of the geometry of the universal Grassmann manifold is presented along the line sketched by Sato and Sato [32]. General issues underlying the notion of infinite-dimensional manifolds are also discussed. A particular choice of affine coordinates on Grassmann manifolds, for both the finite- and infinite-dimensional case, turns out to be very useful for the understanding of geometric structures therein. The so-called “Kac-Peterson cocycle”, which is physically a kind of “commutator anomaly”, then arises as a cocycle of a Lie algebra of infinitesimal transformations on the universal Grassmann manifold. The action of group elements for that Lie algebra is also discussed. These ideas are extended to a multi-component theory. A simple application to a non-linear realization of current and Virasoro algebras is presented for illustration.

scholarly journals Smoothing operators and C∗-algebras for infinite dimensional Lie groups

2017 ◽  
Vol 28 (05) ◽  
pp. 1750042 ◽  
Author(s):  
Karl-Hermann Neeb ◽  
Hadi Salmasian ◽  
Christoph Zellner
Keyword(s):  
Lie Groups ◽  
Bounded Operator ◽  
Representation Formula ◽  
Unitary Representations ◽  
Direct Integral ◽  
Infinite Dimensional ◽  
C Algebras ◽  
Infinite Dimensional Lie Group ◽  
The One ◽  
Infinite Dimensional Lie Groups

A smoothing operator for a unitary representation [Formula: see text] of a (possibly infinite dimensional) Lie group [Formula: see text] is a bounded operator [Formula: see text] whose range is contained in the space [Formula: see text] of smooth vectors of [Formula: see text]. Our first main result characterizes smoothing operators for Fréchet–Lie groups as those for which the orbit map [Formula: see text] is smooth. For unitary representations [Formula: see text] which are semibounded, i.e. there exists an element [Formula: see text] such that all operators [Formula: see text] from the derived representation, for [Formula: see text] in a neighborhood of [Formula: see text], are uniformly bounded from above, we show that [Formula: see text] coincides with the space of smooth vectors for the one-parameter group [Formula: see text]. As the main application of our results on smoothing operators, we present a new approach to host [Formula: see text]-algebras for infinite dimensional Lie groups, i.e. [Formula: see text]-algebras whose representations are in one-to-one correspondence with certain continuous unitary representations of [Formula: see text]. We show that smoothing operators can be used to obtain host algebras and that the class of semibounded representations can be covered completely by host algebras. In particular, the latter class permits direct integral decompositions.

scholarly journals Joseph Fourier 250thBirthday: Modern Fourier Analysis and Fourier Heat Equation in Information Sciences for the XXIst Century

Entropy ◽  
2019 ◽  
Vol 21 (3) ◽  
pp. 250
Author(s):  
Frédéric Barbaresco ◽  
Jean-Pierre Gazeau
Keyword(s):  
Fourier Analysis ◽  
Heat Equation ◽  
Harmonic Analysis ◽  
Lie Groups ◽  
Coherent States ◽  
Geometric Theory ◽  
Plancherel Formula ◽  
Group Representations ◽  
Modern Development ◽  
Xxist Century

For the 250th birthday of Joseph Fourier, born in 1768 at Auxerre in France, this MDPI special issue will explore modern topics related to Fourier analysis and Fourier Heat Equation. Fourier analysis, named after Joseph Fourier, addresses classically commutative harmonic analysis. The modern development of Fourier analysis during XXth century has explored the generalization of Fourier and Fourier-Plancherel formula for non-commutative harmonic analysis, applied to locally compact non-Abelian groups. In parallel, the theory of coherent states and wavelets has been generalized over Lie groups (by associating coherent states to group representations that are square integrable over a homogeneous space). The name of Joseph Fourier is also inseparable from the study of mathematics of heat. Modern research on Heat equation explores geometric extension of classical diffusion equation on Riemannian, sub-Riemannian manifolds, and Lie groups. The heat equation for a general volume form that not necessarily coincides with the Riemannian one is useful in sub-Riemannian geometry, where a canonical volume only exists in certain cases. A new geometric theory of heat is emerging by applying geometric mechanics tools extended for statistical mechanics, for example, the Lie groups thermodynamics.

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